Tracking of lines in spherical images via sub-Riemannian geodesics in SO(3)

A. Mashtakov, R. Duits, Y. Sachkov, E.J. Bekkers, I. Beschastnyi

Research output: Contribution to journalArticleAcademicpeer-review

8 Citations (Scopus)
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Abstract

In order to detect salient lines in spherical images, we consider the problem of minimizing the functional ∫0lC(γ(s))ξ2+kg2(s)ds for a curve γ on a sphere with fixed boundary points and directions. The total length l is free, s denotes the spherical arclength, and k g denotes the geodesic curvature of γ. Here the smooth external cost C≥ δ> 0 is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group SO(3) and show that the spherical projection of certain SR geodesics provides a solution to our curve optimization problem. In fact, this holds only for the geodesics whose spherical projection does not exhibit a cusp. The problem is a spherical extension of a well-known contour perception model, where we extend the model by Boscain and Rossi to the general case ξ> 0 , C≠ 1. For C= 1 , we derive SR geodesics and evaluate the first cusp time. We show that these curves have a simpler expression when they are parameterized by spherical arclength rather than by sub-Riemannian arclength. For case C≠ 1 (data-driven SR geodesics), we solve via a SR Fast Marching method. Finally, we show an experiment of vessel tracking in a spherical image of the retina and study the effect of including the spherical geometry in analysis of vessels curvature.

Original languageEnglish
Pages (from-to)239–264
Number of pages26
JournalJournal of Mathematical Imaging and Vision
Volume58
Issue number2
DOIs
Publication statusPublished - Jun 2017

Keywords

  • Sub-Riemannian Geodesics
  • Lie Group SO(3)
  • Vessel Tracking
  • Geometric Control
  • spherical images
  • Lie group SO(3)
  • Geometric control
  • Sub-Riemannian geodesics
  • Spherical image
  • Vessel tracking

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